Elementary Statistics Final
a494533693
finalexamGUIDE_2.pdf
Chapter 1 ( Definitions) Parameter
Statistic
population mean
x sample mean
population standard deviation
s sample standard deviation
2
population variance
2 s sample variance
P population proportion
p̂ sample proportion
Chapter 2 and 3 (Descriptive Statistics) Find the mean, median, mode, midrange, standard deviation, and variance given raw data.
TI-83/84 Stat, Calc, 1-varstats L1 gives you most of this information (L1 is where you
entered your data)
Find the mean (weighted mean), median, mode, standard deviation, and variance in a
frequency table. TI-83/84 Stat, Calc, 1-varstats L1,L2 gives you most of this
information (L1 is your class midpoints and L2 is your frequency)
You can get your variance by squaring the unrounded standard deviation. After the
1-varstats go to VARS, statistics and scroll down to select Sx and press enter. Select the 2
x button to square the unrounded standard deviation and press enter.
Find relative frequency, cumulative frequency, class boundaries, class midpoints, class
width, upper and lower class limits from a frequency table.
Construct a histogram, frequency polygon, pie chart,…
Know how to use the Empirical Rule
Chapter 4 (Probability) 0 P(A) 1
P(A) + P( A ) = 1
Find probability of A or B
P(A or B) = P(A) + P(B) if the events are mutually exclusive
P(A or B) = P(A) + P(B) - P(A and B) if the events are not mutually exclusive
Find probability of A and B
If A and B are dependent:
If A and B are independent:
( ) ( ) ( | )P A and B P A P B A
)()()( BPAPBandAP
Find probability of B given A
( and ) ( | )
( )
P A B P B A
P A
Find the probability of “at least one”
P(at least one) = 1 – P(none)
Know what’s in a deck of cards as I will use it to ask you probability questions.
Review all probability questions asked in homework, quizzes, and exams.
Counting rules:
Permutations
TI 83/84: enter your value for n, MATH, right arrow to PRB, scroll down to nPr, press
enter, then enter the value for r, press enter.
Combinations:
TI 83/84: enter your value for n, MATH, right arrow to PRB, scroll down to nCr, press
enter, then enter the value for r, press enter.
Chapter 5 (Probability Distributions)
Find expected values ( )E x p x know what it means for E = 0, E > 0, and E < 0
Find the mean of a probability distribution ( )x p x
Find the standard deviation of a probability distribution 2 2
( )x p x
Find the VARIANCE of a probability distribution 2
Find the missing value in a probability distribution table. Remember that ( ) 1p x
TI-83/84: Finding mean and standard deviation of a probability distribution table
Enter x into L1 and P(x) into L2 then go to STAT, CALC and select 1-varstats
L1,L2 ( is the mean and is the standard deviation X
X )
)!(
!
rn
n P
rn
)!(!
!
rnr
n C
rn
(Binomial Probability Distributions) Find the mean of a BINOMIAL probability distribution np
Find the standard deviation of a BINOMIAL probability distribution npq
Find the VARIANCE of a BINOMIAL probability distribution 2
Keywords: “exactly”, “at least”, “at most”
You will be asked to find the probability of exactly, at least or at most. You can use your
TI-83/84, the binomial tables or your binomial formula.
Example if you are using the TI -83/84:
Given n = 10, p =.25
a) find probability of exactly 3 use BinomPDF(10, .25, 3)
b) find probability of at least 3 use 1-BinomCDF(10, .25, 2)
c) find the probability of at most 3 use BinomCDF(10, .25, 3)
Note: at least 3 means x = 3,4,5,6,7,8,9,10 at most 3 means x = 0,1,2,3
To get BinomPDF or BinomCDF you need to go to 2 nd
VARS and scroll down.
Chapter 6 (Normal Distributions) Area under the curve represents probability
Given the mean and standard deviation you are asked to find the probability.
Use x
z
then go to table A-2 to find area under the curve
TI-83/84: 2 nd
VARS select normalCDF(left z, right z) also gives probability
Given the mean, standard deviation, and sample size "n" you are asked to find the probability.
use x
z
n
then go to table A-2 to find area under the curve
TI-83/84: 2 nd
VARS select normalCDF(left z, right z) also gives probability
When asked to find the value that separates the top __% from the bottom __%
Use ( )x z
The bottom % represents the left area that will give you z (use table A-2 )
TI-84: 2 nd
VARS select invnorm(area to left) also gives z value. Take this value and plug
it into the formula above.
Chapter 7 (Confidence Intervals – one population) Remember that confidence intervals have two tails
Wording Parameter Formula used critical value
TI 84 select STAT then TESTS
find a ___% confidence interval for the population
mean ( known)
( known)
2
x E x E
E z n
Table A-2 Z-interval
find a ___% confidence interval for the population
mean ( not known)
( unknown)
2
x E x E
s E t
n
Table A-3 T-interval
find a ___% confidence interval for the population standard deviation
2 2
2 2
( 1) ( 1)
R L
n s n s
Table A-4 --------
find a ___% confidence interval for the population proportion
P
2
ˆ ˆ
ˆ ˆ
p E P p E
pq E z
n
Table A-2 1-propZint
Common Critical values: Confidence Intervals Critical Value
.90 1.645
.95 1.96
.99 2.575
.98 2.33
Sample Size Determination: Find the sample size needed to ……
2
2
2
( ) (0.25)z n
E
Use when , ˆ ˆand p q are
not given
2
2
2
ˆ ˆ( )z pq n
E
Use when ˆ ˆand p q are
given
2
2 z
n E
Use when is given
When finding sample size ALWAYS round up. Example: n = 134.01 would be n = 135
Chapter 8 (Hypothesis testing – one population) Wording: "test the claim that…." 1) The hypothesis is broken into 2 parts H0 - null hypothesis H1 - alternate hypothesis If the claim has the word “is” then it goes in the H0 If the claim has the words “greater than”, “less than”, “different from” then it goes in the H1 It is important where you put the claim because you will be coming back to this as you are deciding on how to word your final conclusion.
WORDING SYMBOL
IS = ( ALWAYS IN THE HO)
DIFFERENT FROM GREATER THAN >
LESS THAN <
One of three population parameters will be tested. The population parameters are mean (),
standard deviation ( ), and proportion (P)
2) Calculate the test statistic There are 4 test statistics - the population parameter being tested determines which test statistic you will use.
PARAMETER BEING TESTED TEST STATISTIC USED TI-84 select STAT, TESTS
( known) z
x
n
Z-test
( unknown) t x
s n
T-test
2
(n 1)s 2
2
-------------
P z
p̂ p
pq n
1-propZtest
3) Find your Critical Region Determine how many tails you are working with. Is it a two tailed, left tailed or right tailed? This depends on your set up in the H1 (alternate hypothesis) Example:
H0: = 3
H1: 3
H0: = 3
H1: >3
H0: = 3
H1: <3
means you will be
working with TWO TAILS
> means you will be working with A RIGHT TAIL
< means you will be working with A LEFT TAIL
Remember that the significance level represents the area in the shaded region. After you determine the critical region, find the critical values. Critical values separate the critical region from the non-critical region. Then you will see where your test statistic falls in comparison to the critical values. Finding critical values: You need to decide which table you will be using. Is it A-2, A-3 or A-4? The table below should help in making that decision.
PARAMETER BEING TESTED
TEST STATISTIC USED
USING OBTAIN CRITICAL VALUES FROM THE GIVEN TABLE
( known) z
x
n
A-2 Invnorm of left area also gives you your critical value
( unknown) t x
s n
A-3
2
(n 1)s 2
2
A-4
P z
p̂ p
pq n
A-2 Invnorm of left area also gives you your critical value
You will be using the significance level and the appropriate table to find your critical values.
Two tails – divide
significance level by 2
Right tail
Left tail
In all cases, the shaded
region is known as the
CRITICAL REGION. If
your test statistic falls in the
critical region then you will
REJECT the H0
4) Conclusion In your book you will find a chart that will help you with your conclusion. It starts by asking if the original claim contains equality. Does your claim have the “=” symbol? If so it is in the H0 . If it does not have the “=” symbol then it’s in the H1. Continue with the chart to get the CORRECT conclusion. Remember that you will either reject H0 or fail to reject H0 H0 is rejected when the test statistic falls within the critical region. ----------------------------------------------------------------------------------------------------
USING THE P-VALUE METHOD:
1) Set up the hypothesis H0 - null hypothesis
H1 - alternate hypothesis
2) Test statistic 3) Find the p-value
4) Conclusion
H0 is rejected when the p-value . That means that the p-
value has to be less than or equal to the significance level.
5) Wording of final conclusion: Write your conclusion in non
technical terms - use the chart in your book
Use your calculator to find the p-value. Follow the chart below.
PARAMETER BEING TESTED
CALCULATOR COMMAND to find p-value
( known) Stat, tests, z-test
( unknown) Stat, tests, t-test **
*See note below
P Stat, tests, 1propz-test
* Other ways to find p-values when you know the test statistic ( 2
and t)
RIGHT TAILED TEST: . 2nd, VARS, 2 cdf ( ), x2cdf(x2,E99,n-1) note: to get E press 2
nd , EE
LEFT TAILED TEST: . 2nd, VARS, 2 cdf ( ), x2cdf(0,x2,n-1)
TWO TAILED TEST: take the smallest of the two above and multiply by 2 RIGHT TAILED TEST: 2nd, vars, tcdf(t,E99,n-1) LEFT TAILED TEST: 2nd, vars, tcdf(-E99,t,n-1) TWO TAILED TEST: 2nd, vars, the answer for the right tailed test and multiply it by two
Chapter 9 (Hypothesis testing – two populations) “test the claim that….”
PARAMETER BEING TESTED
TEST STATISTIC USED TI-84 select STAT, TESTS
Two
Proportions 1 2
,P P 1 2 1 2
1 2
ˆ ˆ( ) ( )p p p p z
pq pq
n n
2-propZtest
Two Means –
independent (
two different
groups)
1 2 ,
1 2 1 2
2 2
1 2
1 2
1 2
1 2
( ) ( ) where
df = smaller of -1 or -1
and unknown and not assumed equal
x x t
s s
n n
n n
2-sampTtest
Two Means –
matched pairs
( same group
before and
after)
d where d = x-
y
X = before
Y = after
where
df = 1
d
d
d t
s
n
n
T-test ( on
the
differences
“d”)
Chapter 9 (Confidence Intervals – two populations) "Construct a ___% confidence interval for the…"
Confidence interval for
Formula USED TI-84 select STAT, TESTS
Two
Proportions 1 2
P P 1 2 1 2 1 2
1 1 2 2
2 1 2
ˆ ˆ ˆ ˆ( ) ( ) where
ˆ ˆ ˆ ˆ
p p E p p p p E
p q p q E z
n n
2-propZint
Two Means
–
independent (
two different
groups)
1 2
1 2 1 2 1 2
2 2
1 2 1 2
2 1 2
1 2
( ) ( ) ( ) where
E df = smaller of -1 or -1
and unknown and not assumed equal
x x E x x E
s s t n n
n n
2-sampTint
Two Means
– matched
pairs ( same
group before
and after)
d where d
= x-y
X = before
Y = after 2
where
df = 1
d
d
d E d E
s E t
n
n
T-interval (
on the
differences
“d”)
If zero is included in your confidence interval then this indicates that there is no difference between the two
groups. For Example: -0.25 < 1 2 < 0.54 You can see that zero is included in the interval this means that
1 2 = 0 which indicates that
1 2
Chapter 10 (linear regression) Find the linear correlation coefficient (r)
Determine if a significant linear correlation exists
Find the best predicted ŷ when x is given
If there is a significant linear correlation then use the regression equation to make predictions.
If there is NO significant linear correlation then use y to make predictions
TI83/84 Instructions:
1. Hit Stat, Edit.
2. Enter your data into any two lists, preferably L1 and L2 since they are the
default.
3. To create a scatter plot, we need to get into Stat-Plot, which is above the Y= key,
the upper left hand button.
4. Once in Stat-Plot, we select the first plot, highlight On and hit enter if it is not
already turned on, select the first type of plot from the six available, make sure
L1 and L2 are the x and y lists unless your data is in another set of lists, and then
select the mark we want used.
5. Now, we hit Zoom, which is in the middle of the top buttons, and select the 9 th
option-Zoom Stat. This will bring up our scatter plot, it ZOOMs in on the
STATistical data.
6. If it says Dim Mismatch or some such error, look at your lists, there may be one
more entry in one list than the other, so the DIMensions aren’t the same. Or, look
in the Y= area. If there are any equations in any of the “y=” spots, delete them.
7. Now, to find the line of best fit and correlation coefficient information, we hit
Stat, Calc, 8:LinReg (a+bx). This will bring up what a=, b=, r squared, and r.
(*If r doesn’t show up, then hit 2 nd
, Catalog (above 0), D, DiagnosticsOn, enter,
enter.*)
8. Once you have the line of best fit, you can enter it into Y= and hit graph to see it
fitted onto your data. If it doesn’t seem to fit the data, a mistake has occurred
somewhere, go find it.
Chapter 11
Testing for independence… STAT, Tests, 2
- Test
H0: one variable INDEPENDENT of second variable H1: one variable DEPENDENT of second variable
Test Statistic:
2 2 ( )O E
E
or use TI 83/84
TI 83/84 instructions:
1) 2 nd
, 1
x
( on some calculators press MATRIX )
2) Right arrow to EDIT press enter 3) Enter your observed values in matrix and press 2
nd QUIT when done
4) Press STAT and right arrow to TEST
5) Select 2
- Test then press enter 6) You will see that your expected values are stored in matrix B and your observed values are stored
in matrix A. Select calculate at the bottom of your screen and press enter.
7) You should now see your tests statistic and p-value.
8) If you want to see your expected value, go to matrix B. 2 nd
, 1
x
( on some calculators press
MATRIX ) select B and press enter twice.
Critical Value: Always right tail. Obtain from table A-4
Goodness-of-Fit Tests….. (with one row of data)
H0: all probabilities equal ( or equal to some claimed amount) H1: at least one is different from the others
Test Statistic:
2 2 ( )O E
E
If you wish to use your TI83/84:
1) Enter the observed values (O) into L1 2) Calculate E: Use E = n/k ( if all frequencies are equal) and use E = n*p if all frequencies are
not equal. Enter E into L2.
3) Now go to STAT, Tests, “ 2
- GOF Test” where L1 = Observed and L2 = Expected. If
your calculator doesn’t have this command then you can do the following: sum(( L
1 E)
2 / E)
and press enter to get the test statistic. “sum” is under 2 nd
STAT, right arrow to “math” it’s the 5 th
option. Now press to select it.
Critical Value: Always right tail. Obtain from table A-4
Testing that 3 or more means are equal: STAT, Tests, ANOVA
H0: µ1 = µ2 = µ3 = . . . H1: at least one is different from the others Enter your data into L1, L2, L3,…. So that it looks like this Anova(L1,L2,L3) press enter.
Now you have your test statistics and p-value
